Math Problems Vault

Problem 5.138 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

Let

f(z) = a_0+\binom {n}{1}a_1z+\binom {n}{2}a_2z^2+\cdots +\binom {n}{n-1}a_{n-1}z^{n-1}+a_nz^n

be an arbitrary polynomial of degree $n$ and let $\zeta_1, \zeta_2, \ldots , \zeta_n$ be arbitrary points in the complex plane. The expression

A(\zeta _1, \zeta _2, \ldots , \zeta _n)f(z) = \frac{1}{n!}A_{\zeta _1}A_{\zeta _2}\ldots A_{\zeta _n}f(z)

is a symmetric multilinear function of the variables $\zeta_1, \zeta_2, \ldots , \zeta_n$ , which we can write as follows:

A(\zeta _1, \zeta _2, \ldots , \zeta _n)f(z) = a_0\Sigma _0+a_1\Sigma _1+a_2\Sigma _2+\cdots +a_{n-1}\Sigma _{n-1}+a_n\Sigma _n.

Here $\Sigma_0, \Sigma_1, \Sigma_2, \ldots , \Sigma_{n-1}, \Sigma_n$ are the elementary symmetric functions of $\zeta_1, \zeta_2, \ldots , \zeta_n$ :

\Sigma _0=1, \qquad \Sigma _1=\zeta _1+\zeta _2+\cdots +\zeta _n,

\Sigma _2=\zeta _1\zeta _2+\zeta _1\zeta _3+\cdots +\zeta _{n-1}\zeta _n, \qquad \ldots , \qquad \Sigma _n=\zeta _1\zeta _2\ldots \zeta _n.

If the last $k$ of the numbers $\zeta_1, \zeta_2, \ldots , \zeta_n$ (and only these) lie at infinity, then we set $\Sigma_0=\Sigma_1=\cdots =\Sigma_{k-1}=0$ , $\Sigma_k=1$ , $\Sigma_{k+1}=\zeta_1+\zeta_2+\cdots +\zeta_{n-k}$ , $\Sigma_{k+2}=\zeta_1\zeta_2+\zeta_1\zeta_3+\cdots +\zeta_{n-k-1}\zeta_{n-k}, \ldots , \Sigma_n=\zeta_1\zeta_2\ldots \zeta_{n-k}$ .

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Problem 5.139 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

Let

f(z) = a_0+\binom {n}{1}a_1z+\binom {n}{2}a_2z^2+\cdots +\binom {n}{n-1}a_{n-1}z^{n-1}+a_nz^n

be an arbitrary polynomial of degree $n$ with zeros $z_1, z_2, \ldots , z_n$ and

g(z) = b_0+\binom {n}{1}b_1z+\binom {n}{2}b_2z^2+\cdots +\binom {n}{n-1}b_{n-1}z^{n-1}+b_nz^n

an arbitrary polynomial of degree $n$ with zeros $\zeta_1, \zeta_2, \ldots , \zeta_n$ . Calculate $A(\zeta_1, \zeta_2, \ldots , \zeta_n)f(z)$ and $A(z_1, z_2, \ldots , z_n)g(z)$ .

If the two polynomials are related in such a manner that

A(\zeta _1, \zeta _2, \ldots , \zeta _n)f(z) = 0

or, equivalently [solution 139], $A(z_1, z_2, \ldots , z_n)g(z)=0$ , then the two polynomials $f(z)$ and $g(z)$ are termed apolar. The term suggests the vanishing of the $n$ th polar $A_{\zeta_1}A_{\zeta_2}\ldots A_{\zeta_n}f(z)$ . The condition of apolarity is

a_0b_n-\binom {n}{1}a_1b_{n-1}+\binom {n}{2}a_2b_{n-2}-\cdots +(-1)^{n-1}\binom {n}{n-1}a_{n-1}b_1+(-1)^na_nb_0=0.

[139.] The two systems $z_1, z_2, \ldots , z_n$ and $\zeta_1, \zeta_2, \ldots , \zeta_n$ are also termed mutually apolar.

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Problem 5.140 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

What is the geometrical significance of the apolarity of $z_1, \zeta_1$ , and that of $z_1, z_2$ and $\zeta_1, \zeta_2$ ?

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Problem 5.141 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

Which are the systems $\zeta_1, \zeta_2, \ldots , \zeta_n$ , that are apolar to the system consisting of the roots of the equation $z^n-1=0$ ?

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Problem 5.142 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

Let $z_1, z_2, \ldots , z_n$ be an arbitrary system of points. There are $n$ systems $\zeta ,\zeta ,\ldots ,\zeta$ of coincident points that are apolar to the given system. Which are they?

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Problem 5.143 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

The polynomial of degree $n$

f(z) = 1-z+cz^n

is apolar to any system of numbers whose sum is $n$ and product is $0$ .

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Problem 5.145 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

Let the two polynomials $f(z)$ and $g(z)$ of degree $n$ be apolar to each other. Every circular domain that contains all the zeros of one polynomial also contains at least one zero of the other.

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Problem 5.146 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

Let the two polynomials $f(z)$ and $g(z)$ of degree $n$ be apolar to each other. The two smallest convex polygons that contain all the zeros of $f(z)$ and $g(z)$ respectively have at least one common point. More generally, the two smallest circular-arc polygons that are circularly convex with respect to the same point and that contain all the zeros of $f(z)$ and $g(z)$ respectively, also have at least one common point.

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Problem 5.147 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

The polynomial

1-z+cz^n

always has one zero in the circle $\left|z\right| \leq 2$ .

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Problem 5.148 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

The polynomial

1-z+cz^n

always has one zero in the circle $\left|z-1\right| \leq 1$ .

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Problem 5.149 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

The polynomial of $k+1$ terms

1-z+c_2z^{\nu _2}+c_3z^{\nu _3}+\cdots +c_kz^{\nu _k}, \qquad 1=\nu _1<\nu _2<\nu _3<\cdots <\nu _k

always has one zero in the circle

\left|z\right| \leq \left[\left(1-\frac{1}{\nu _2}\right)\left(1-\frac{1}{\nu _3}\right)\cdots \left(1-\frac{1}{\nu _k}\right)\right]^{-1},

and hence in the circle

\left|z\right| \leq k.

(Generalization of 147.)

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Problem 5.150 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Derivative of a Polynomial with respect to a Point. A Theorem of Grace"

If a polynomial of degree $n$ assumes the same value at two points $a$ and $b$ , $a \neq b$ , then its derivative has at least one zero in the circular disc with center at the midpoint of the line segment $ab$ and radius $(\left|a-b\right|/2)\cot (\pi /n)$ . (Analogue of Rolle's theorem in the complex domain.)

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Grace's theorem on apolar polynomials

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